let p, q be Element of CQC-WFF ; :: thesis: for x being bound_QC-variable holds (All x,(p '&' q)) => ((All x,p) '&' q) is valid
let x be bound_QC-variable; :: thesis: (All x,(p '&' q)) => ((All x,p) '&' q) is valid
A1: All x,((p '&' q) => (q '&' p)) is valid by Th26, CQC_THE1:103;
(All x,((p '&' q) => (q '&' p))) => ((All x,(p '&' q)) => (All x,(q '&' p))) is valid by Th34;
then A2: (All x,(p '&' q)) => (All x,(q '&' p)) is valid by A1, CQC_THE1:104;
(All x,(q '&' p)) => (q '&' (All x,p)) is valid by Th68;
then A3: (All x,(p '&' q)) => (q '&' (All x,p)) is valid by A2, LUKASI_1:43;
(q '&' (All x,p)) => ((All x,p) '&' q) is valid by CQC_THE1:103;
hence (All x,(p '&' q)) => ((All x,p) '&' q) is valid by A3, LUKASI_1:43; :: thesis: verum